Das and Kempe [1]}, [2]} consider the problem of maximizing \(R^2_{Z,\mathcal {S}}\)  subject to the constraint \(\left|\mathcal {S}\right| \le k\) . They showed that although the \(R^2\)  objective is not submodular in general, it is submodular in the absence of suppressors [1]}.
Moreover, they motivate the use of the greedy algorithm in practice by showing that the \(R^2\)  objective satisfies (REF ) for all \(\mathcal {A}\subseteq \mathcal {B}\subseteq \mathcal {X}\)  with \(\gamma \ge \lambda _{\min }(C_{\mathcal {X}}, \left|\mathcal {B}\right|) \ge \lambda _{\min }(C_{\mathcal {X}})\)  is a \(\lambda _{\min }(C_{\mathcal {X}})\)  is the smallest eigenvalue of \(C_{\mathcal {X}}\)  and \(\lambda _{\min }(C_{\mathcal {X}}, \left|\mathcal {B}\right|)\)  is the smallest \(\left|\mathcal {B}\right|\) -sparse eigenvalue of \(C_{\mathcal {X}}\) [2]}. In order to analyze the greedy algorithm, it suffices to let \(\mathcal {B}\setminus \mathcal {A}\)  and \(\mathcal {A}\)  both be subsets containing at most \(k\)  variables, so \(\left|\mathcal {B}\right| = 2k\) .
